Designing Decentralized Control Systems without Structural Fixed Modes: A Multilayer Approach ⋆ S´ ergio Pequito 1,2 Clarance Agbi 1 Nipun Popli 1 Soummya Kar 1 A. Pedro Aguiar 2,3 Marija Ili´ c 1 1 Department of Electrical and Computer Engineering, Carnegie Mellon University, Pittsburgh, PA 15213. 2 Institute for System and Robotics, Instituto Superior T´ ecnico, Technical University of Lisbon, Lisbon, Portugal. 3 Faculty of Engineering, University of Porto (FEUP), Portugal. Abstract: In this paper we propose a methodology to design decentralized controllers applied to large-scale systems. The key idea is to split the design into two control layers. The first layer, consists mainly in a pre-processing step, where an optimal subset of inputs and outputs are used for feedback to close the loop, such that the resulting associated state matrix of the closed- loop system has two desirable structural properties: it has no structural fixed modes and it is structurally controllable and observable thru any single state variable. After this first layer, we can select an arbitrary decentralization control scheme to achieve some specified performance, which is the second layer of the proposed approach. We illustrate this methodology for the following applications: a two zones temperature regulation and a frequency regulation of a 3- bus system. Keywords: Structural Systems, Structural Fixed Modes, Decentralized Control 1. INTRODUCTION In geographically distributed large-scale systems such as power systems, multi-agent networks, cyber-physical sys- tems, to name a few, a decentralized control structure is often more desirable than a centralized one. This is due the fact that typically, it is not realistic to assume that each control input signal can be generated by using all the measurement signals of the system. In other words, some kind of constraint on the information structure is in- evitable. Decentralized control theory has attracted several researchers in the past decades, and several methodologies have been proposed, see for instance Sandell et al. (1978); Bakule and Lunze (1988); Siljak (1991, 2007). In this paper we consider a possible large scale plant modeled by ˙ x = Ax, where x ∈ R n is the state of the system. The goal is to design the input matrix structure ¯ B, the output matrix structure ¯ C and a constrained information pattern ¯ K such that typical decentralized control methods hold. To this end, we propose a multilayer approach that is composed by two layers. The first layer of our approach consists in the following ⋆ This work was partially supported by grant SFRH/BD/33779/2009, from Funda¸c˜ao para a Ciˆ encia e a Tecnologia (FCT) and the CMU-Portugal (ICTI) program, and by projects CONAV/FCT-PT (PTDC/EEACRO/113820/2009), FCT (PEst-OE/EEI/LA0009/2011), and MORPH (EU FP7 No. 288704). E-mail: {spequito,cagbi,npopli,soummyak}@andrew.cmu.edu, pedro.aguiar@fe.up.pt, milic@ece.cmu.edu L1 Consider the structural pattern ¯ A of A (i.e., the zero/non-zero pattern). In this first layer, the goal is to find the structural matrices ( ¯ B 1 , ¯ K 1 , ¯ C 1 ) such that the associated state matrix of the closed-loop system given by A 1 = A + B 1 K 1 C 1 , with a numerical realization 1 (B 1 ,K 1 ,C 1 ) with the same structural pattern as ( ¯ B 1 , ¯ K 1 , ¯ C 1 ) satisfies the following: the closed-loop system A 1 composed by ( ¯ A, ¯ B 1 , ¯ K 1 , ¯ C 1 ) has no structural fixed modes (to be made precise in Section II) and is such that it is structurally controllable 2 (resp. observable) by ma- nipulating (resp. measuring) any single state variable. Note that at the end of this first control layer, we obtain a matrix A 1 that (by construction) satisfies the commonly required conditions to perform decentralized control once we consider an arbitrary set of inputs, outputs and feed- back links. Hence, we can design a controller to achieve some performance. This is done in the next control layer. 1 Note that the parameters used by (B 1 ,C 1 ) are imposed by physical constraints or modelling assumptions, whereas K 1 is determined by a parametric choice of our interest. 2 A pair (A, B) is said to be structurally controllable if there exists a pair (A ′ ,B ′ ) with the same structure as (A, B), i.e., same locations of zeroes and non-zeroes, such that (A ′ ,B ′ ) is controllable. By density arguments, it may be shown that if a pair (A, B) is structurally controllable, then almost all (with respect to the Lebesgue measure) pairs with the same structure as (A, B) are controllable. In essence, structural controllability is a property of the structure of the pair (A, B) and not the specific numerical values. A similar definition and characterization holds for structural observability (with obvious modifications).